Suppose that $W$ is a linear subspace of $\mathbb R^n$. Does it trivially follow that $W$ is a regular submanifold of $\mathbb R^n$?
I would take open balls and the identity function restricted to them composed with a suitable rotation as the charts satisfying the regular submanifold property. Is it correct?
(Definition: Suppose that $M$ is a smooth manifold of dimension $m$, and $N$ is a topological subspace. $N$ is called a smooth regular submanifold of $M$ provided that for each $x\in N$ there exists a chart $(U,\psi)$ in the maximal atlas for $M$ with $x\in U$ such that $\psi(U\cap N)=\psi(U) \cap (\mathbb R^n \times {0_{\mathbb R^{m-n}}})$ for some $n\le m$.)